Transitions and anti-integrable limits for multi-hole Sturmian systems and Denjoy counterexamples

For a Denjoy homeomorphism f of the circle S , we call a pair of distinct points of the ω -limit set ω( f ) whose forward and backward orbits converge together a gap , and call an orbit of gaps a hole . In this paper, we generalize the Sturmian system of Morse and Hedlund and show that the dynamics of any Denjoy minimal set of finite number of holes is conjugate to a generalized Sturmian system. Moreover, for any Denjoy homeomorphism f having a finite number of holes and for any transitive orientation-preserving homeomorphism f 1 of the circle with the same rotation number ρ( f 1 ) as ρ( f ) , we construct a family f ε of Denjoy homeomorphisms of rotation number ρ( f ) containing f such that (ω( f ε ) , f ε ) is conjugate to (ω( f ) , f ) for 0 < ε < ˜ ε < 1, but the number of holes changes at ε = ˜ ε , that (ω( f ε ) , f ε ) is conjugate to (ω( f ˜ ε ) , f ˜ ε ) for ˜ ε (cid:2) ε < 1 but lim ε (cid:3) 1 f ε ( t ) = f 1 ( t ) for any t ∈ S , and that f ε has a singular limit when ε (cid:3) 0. We show this singular limit is an anti-integrable limit (AI-limit) in the sense of Aubry. That is, the Denjoy minimal system reduces to a symbolic dynamical system. The AI-limit can be degenerate or nondegenerate. All transitions can be precisely described in terms of the generalized Sturmian systems.


Introduction
Let f be a continuous map of a topological space X. We say that a subset Y of X is invariant under f if f (Y) = Y. A closed invariant set Y is called minimal if it contains no proper closed invariant subsets. Let S = {z ∈ C||z| = 1} be the unit circle. We identify S with R/Z and have the identification [0, 1) t with {z ∈ C||z| = 1} e 2π it . We shall freely use the representation of S that is most convenient. Let β ∈ (0, 1) and R β : S → S, t → t + β(mod1), be the rotation with angle β.
For a monotone twist map of the cylinder S × R, the celebrated Aubry-Mather theory tells that we can always find invariant minimal closed subsets of the cylinder on which the twist map has irrational rotation numbers. These closed subsets are either Lipschitz circles or Cantor sets on Lipschitz circles. In the latter case, they are also called cantori or Denjoy minimal sets. The projection of an invariant circle or a cantorus to the unit circle S is the circle S or a Cantor set C in the circle, respectively. And, the restriction of the twist map to an invariant circle or a cantorus projects to a homeomorphism of S or C , respectively. In the latter case, we can extend the homeomorphism linearly into the complement S \ C to obtain a circle homeomorphism. The rotation number of an invariant circle or a cantorus is defined to be the rotation number of this induced circle homeomorphism. (See (2.1) in Section 2 for the definition of rotation number of a circle homeomorphism.) 2 Y.-C. CHEN The Aubry-Mather theory also indicates that an invariant circle breaks by the conjugacy from an irrational rotation becoming discontinuous. (If f 1 and f 2 are respectively maps of spaces X 1 and X 2 , and Y 1 ⊆ X 1 , Y 2 ⊆ X 2 are invariant sets of f 1 and f 2 , respectively. The restriction of f 1 to Y 1 is said to be (topologically) semi-conjugate to the restriction of f 2 to Y 2 if there exists a continuous surjection h : Y 1 → Y 2 , called a semi-conjugacy, such that f 2 • h = h • f 1 on Y 1 . If h is a homeomorphism, then it is called a conjugacy.) For a large class of maps of the cylinder and rotation numbers, the breakup boundary in parameter space is believed to be smooth. But for the two-harmonic family of maps of the following form (1.1) of S×R with parameters a and b (and for multiharmonic maps in general), the breakup boundary exhibits a fractal structure. Baesens & MacKay (1993) believe that this is because cantori of this family of fixed rotation number form an interval in the vague topology, and thus the breakup boundary is composed of many pieces, each one corresponding to a point in the interval. (Each cantorus carries a unique invariant measure, induced by semi-conjugacy to rotation, and is the support of that measure. Suppose μ 1 and μ 2 are the invariant measures on two cantori. The vague topology on cantori is defined by saying that the two cantori are close if the integrals of any continuous function of compact support with respect to μ 1 and μ 2 are close. See Mather (1985) for more details about the vague topology.) Baesens & MacKay (1993) (see also MacKay, 1992) showed that for maps of the form (1.1) and (1.2) near enough an anti-integrable limit (AI-limit) (to be explained shortly) cantori of a given rotation number may form an interval (in the vague topology). This is because the cantori may have multiple number of holes. Following their terminology, we call a pair of distinct points of a Denjoy minimal set whose forward and backward orbits converge together a gap. The gaps come in orbits. We call an orbit of gaps a hole. Aubry calls it a discontinuity class (see Aubry et al., 1991). For the two-harmonic family (1.1) and (1.2), the cantori depend on parameters a and b. Baesens and MacKay proved that there are parameter regimes such that on passing different regimes there exists a bifurcation in which a one-hole cantorus gains a second hole or there exists an invariant circle to one-hole cantorus transition. See also Baesens & MacKay (1994) for numerical demonstration.
Note that for an area-preserving monotone twist map, Mather (1985) showed that if there is no invariant circle of a given irrational rotation number β, then there exist uncountably many Denjoy minimal sets of that rotation number. Moreover, as pointed out by Boyland (1987), these are n-fold Denjoy minimal sets, i.e., they wrap n-times around the cylinder, with average speed β for all n loops with n 2. The n-fold Denjoy minimal sets showed by Mather have dimension n − 1 in the vague topology.
A dynamical systems is, in Aubry's sense (see Aubry & Abramovici, 1990), at the AI-limit if it becomes nondeterministic and reduces to a subshift of finite type. For more details on the concept of AI-limit and its applications, see e.g., Aubry (1995), Aubry & Abramovici (1990), Baesens et al. (2013), Chen (2010) and MacKay & Meiss (1992). For the family of maps of the form (1.1) and (1.2), the AI-limit corresponds to the limit a, b → ∞. The orbits of this family of maps are equivalent to those of the following family of recurrence relations:  3 If we set a, b → ∞ with a/b = κ = 1/4, as did in Baesens & MacKay (1994), then for each i the recurrence relations above reduce to an algebraic equation which can be solved easily. Let x i = x + or x − (mod1) be the only two solutions of cos 2π x i + 1 4 = 0. Then, the sequences (. . . , x −1 , x 0 , x 1 , . . .) (mod 1) consisting of all possible solutions of the equation are sequences of x + and x − . This results in a symbolic dynamics on two symbols. The theory of AI-limit says that, under some technical conditions, the symbolic dynamics at the AI-limit persists to sufficiently large a = κb. Thus, we obtain chaotic orbits of the maps (1.1) and (1.2).
Denjoy proved by constructing examples that there exist circle diffeomorphisms that have irrational rotation number β but are not conjugate to R β . The ω-limit sets of Denjoy's examples are Cantor sets. We refer to any orientation-preserving homeomorphism (OPH) of S with irrational rotation number that is not conjugate to a rotation as a Denjoy homeomorphism or Denjoy counterexample. (Given a circle homeomorphism f and a point x ∈ S, define a set ω( There are circle diffeomorphisms whose Denjoy minimal sets have multiple holes. These diffeomorphisms can be constructed by 'blowing up' points in a multiple number of orbits of R β , instead of only in one orbit. It is natural and interesting to investigate whether similar bifurcation and transition phenomena studied in Baesens & MacKay (1993, 1994 also happen in the minimal sets for Denjoy homeomorphisms of the circle. If they do, can one describe the bifurcations or transitions for multi-hole Denjoy minimal sets in terms of symbolic dynamics? More importantly, what is the AI-limit for Denjoy homeomorphisms? These questions motivated this paper and are the central issues to be addressed. In this paper, we generalize the Sturmian system of Morse & Hedlund (1940) by coding irrational rotations with respect to an arbitrary finite partition on the circle and show that the dynamics of any Denjoy minimal set of finite number of holes is conjugate to a generalized Sturmian system. Notice that it is known (see e.g., Katok & Hasselblatt, 1995) that the restriction of a one-hole Denjoy homeomorphism to its ω-limit set is conjugate to the restriction of the full two-shift homeomorphism to a closed invariant subset. We call a generalized Sturmian system a multi-hole Sturmian system. Moreover, for any Denjoy homeomorphism f having a finite number of holes and for any transitive OPH f 1 of the circle with the same rotation number ρ( f 1 ) as ρ( f ), we construct a one-parameter family f ε of Denjoy homeomorphisms of rotation number ρ( f ) having the following properties. The first property is that The second is that the number of holes changes at ε =ε, corresponding to a transition of cantorus in which a cantorus gains or loses a certain number of holes, and that (ω( f ε ), f ε ) is conjugate to (ω( fε), fε) wheñ ε ε < 1. The third is that lim ε 1 f ε (t) = f 1 (t) pointwisely for any t ∈ S, corresponding to the circle to cantorus transition, and that f ε has a singular limit when ε 0. We show that this singular limit is an AI-limit in the sense of Aubry (1995) (see also Aubry & Abramovici, 1990). That is, the Denjoy minimal set collapses to a set of finite point and the Denjoy minimal system reduces to a symbolic dynamical system. The AI-limit can be degenerate or nondegenerate. All transitions can be precisely described in terms of multi-hole Sturmian sequences.
Roughly speaking, near an AI-limit, (ω( f ε ), f ε ) is conjugate to a multi-hole Sturmian system but reduces to a (topological) factor of that multi-hole Sturmian system at the AI-limit. For instance, suppose 4 Y.-C. CHEN that f ε is constructed by blowing up orbit points R n β (0) of the origin into wandering intervals I (1) n and the length |I (1) n | of I (1) n depends on ε for every n ∈ Z. Then, an AI-limit will correspond to the limit |I (1) n | → 0 for all n except |I (1) 1 | → 1 as ε → 0. Analogously, suppose that f ε is obtained by blowing up points of the two orbits {R n β (0)|n ∈ Z} and {R n β (1/2)|n ∈ Z} into wandering intervals I (1) n and I (2) n , respectively. Again, suppose that the lengths of these intervals depend on ε. Then, a two to one-hole transition of cantorus will occur at ε =ε provided that the length of I (2) n shrinks to zero (the gap corresponding to the boundary of I (2) n is annihilated) for every n when ε =ε but the union n∈Z I (1) n remains constituting the wandering intervals.
The rest of this paper is organized as follows. In the next section, we briefly review fundamental properties of Denjoy homeomorphisms. Before describing a way to code symbolically a Denjoy minimal set in Section 4, we establish in Section 3 the multi-hole Sturmian systems that code irrational rotations with respect to arbitrary partitions on the circle. Section 5 is devoted to the transitions and AI-limits of Denjoy minimal systems. The transitions and AI-limits will be described in terms of quotients of multi-hole Sturmian systems. We postpone all proofs of our theorems until the final section.

Denjoy counterexample
The purpose of this section is twofold. On the one hand, it provides a brief review of well-known facts about Denjoy counterexamples. (For a detailed account, the reader may refer to Cornfeld et al., 1982;Katok & Hasselblatt, 1995;Nitecki, 1971, for instance.) On the other hand, it introduces our assumption on the Denjoy minimal sets to be studied.
Recall that a lift of an OPH f : S → S is a homeomorphism F : R → R that satisfies f (x) = F(x) mod 1 for x ∈ [0, 1) and F(x + 1) = F(x) + 1 for every x ∈ R. Such a lift is unique up to an additive constant: ifF is another lift, thenF(x) = F(x) + m for some integer m. Given a lift, the following limit exists and is independent of x. Define the rotation number ρ( f ) of f by If f is a homeomorphism of S having ω( f ) a Cantor set, then f is semi-conjugate to R β for some irrational number β. In other words, f has R β as a factor. More precisely, there is a unique (up to a rotation) continuous nondecreasing surjection h of degree one such that the diagram below commutes The semi-conjugacy h is one-to-one on S \ n∈Z cl I n , where cl I n denotes the closure of I n .
The topological classification of Denjoy homeomorphisms with a given irrational rotation number β is due to Markley given by a finite or countable collection of orbits of the rotation R β up to a simultaneous translation of all these orbits. For a Denjoy homeomorphism, define We call the number of disjoint orbits of D( f ) under R β the number of holes of ω( f ). The number of holes of a Denjoy minimal set is at least one, and may be infinite. Markley (1970) proved the following.
Theorem 2.1 (Markley 1970). A Denjoy homeomorphism f is semi-conjugate to anotherf via an orientation-preserving surjection if and only if they have the same rotation number and for some 0 α < 1. The surjection is a homeomorphism if and only if equality holds in (2.2) On the other way round, for any given Cantor set C ⊂ S and any countable R β -invariant subset D ⊂ S, one can choose pairwise disjoint open intervals I d , d ∈ D, which have the same cyclic ordering as points in D and whose union d∈D I d is S \ C . Then there exists a continuous surjection h of S such that h −1 (d) = cl I d for all d ∈ D and which is one-to-one on h −1 (S \ D). Moreover, one can construct a homeomorphism f of S with rotation number β so that h satisfies (*), that D( f ) = D and that C is the unique minimal invariant set (equal to ω( f )) under f . Let X be a topological space and f an invertible map of X. Denote the orbit of a point If two OPHsf and f of the circle are conjugate by an orientation preserving (resp. reversing) homeomorphism, then  Markley, 1970). For these reasons, in this paper we concentrate on those Denjoy homeomorphisms of rotation number less than 1/2.
Without loss of generality, we make the following assumption throughout this paper.
Assumption A Let f be a Denjoy homeomorphism. Assume that the number of holes of ω( f ) is finite and equal to some integer K 1. Assume that .
n is a gap. Define the following equivalence relation on ω( f ): for points x, y ∈ ω( f ) and a subsetΘ ⊆ Θ, we say For the sake of convenience of notation, in the sequel, we use The following is well-known.
Theorem 2.3 Let f be a Denjoy homeomorphism satisfying Assumption A. The quotient space

Coding of irrational rotation
First, we describe a way to characterize symbolic codes of an irrational rotation of the unit circle S. It is a generalization of Morse and Hedlund's construction of Sturmian sequences in Morse & Hedlund (1940). Given irrational β ∈ (0, 1/2) and t ∈ S, we investigate the coding of the orbit O(t; R β ) in this section. Let Q ⊂ S be a finite set of real numbers having cardinality N ≥ 2. Suppose Q = {q 1 , q 2 , . . . , q N } with the ordering 0 = q 1 < q 2 < . . . < q N < 1 is a set of N consecutive points on S. We call such a finite set Q a partition set or a set of partition points on the circle S. Partition S into N number of intervals: Denote by (Q) the cardinality of Q. Given a partition set Q, we define Φ = {φ 1 , φ 2 , . . . , φ N } to be a finite real number set of the same cardinality as Q, In other words, ν ± (t; β, Q, Φ) give the itinerary sequences of the orbit of t under R β with respect to the partition Q. We call such a finite set Φ a symbol set or a set of symbols, and call (Q, Φ) a partitionsymbol pair or a pair of partition and symbol sets. Endow the finite set Φ with the discrete topology, and the set of , be the usual shift automorphism. We call the subshift (X β,Q,Φ , σ ) of (Φ Z , σ ) an N-symbol Sturmian system of partition points Q with symbols Φ and rotation number β (where N = (Q) = (Φ)). For the sake of simplicity, (Φ Z , σ ) instead of (Φ Z , σ N ) is used in the rest of this paper provided no ambiguity is caused.
A sequence u ∈ Φ Z is called a rotation sequence if it is a rotation sequence of some partition with some rotation number.
By the definition above, a rotation sequence of partition {0, β} or {0, 1 − β} with irrational rotation number β gives rise to a Sturmian sequence. See subsection 3.1 for a brief account of the Sturmian sequence. We remark that for 0 < c < 1 − β the partition {0, c, c + β/2, 1 − β/2} that divides the circle into four arcs was studied by Hockett & Holmes (1986), but they used two symbols rather than four to characterize a rotation. See also Boyland (1993) for coding rotations with two symbols by more general partitions.
for all 1 i n N = (Q) and n ∈ Z. Hence, with respect to the partition Q, the sequence (u n ) n∈Z is also the itinerary sequence of the orbit of t under the reverse rotation with angle 1 − β.
The minimality of the set X β,Q,Φ means that the set can be defined alternatively to be the orbit closure of any rotation sequence u of partition points Q with symbols Φ and rotation number β. (Actually, we prove the minimality in Theorem 3.2 by showing that (3.2) holds.) For the Sturmian system cases X β, {0,β},{0,1} and X β,{0,1−β},{0,1} , results of Theorem 3.2 were proved in Hedlund (1944).
In virtue of the above theorem, it is necessary that β =β for the two systems to be conjugate. Hence, we shall concentrate on a fixed irrational β and, when no ambiguity is caused, write ν ± (t; Q, Φ) = ν ± (t; β, Q, Φ) and X Q,Φ = X β,Q,Φ to simplify notation.
Given a partition set Q and a symbol set Φ, letQ be a subset of Q. Assume that u, v belong to X Q,Φ . Define the following equivalence relation: It is easy to check that the equivalence relation defined is indeed an equivalence relation. For any two subsetsΘ and Θ of Θ the union ∼Θ ∪ ∼ Θ is again an equivalence relation, and is equal to ∼Θ ∪Θ .
Given a sequence u over a finite alphabet A , the complexity function p = p u : N → N, n → p(n), is defined as the number of distinct words of length n occurring in u. If U is a finite word over A , denote by |U| a the number of occurrences of the letter a ∈ A in U. A sequence u over a two-letter alphabet {0, 1} is called balanced if for any pair of words U, V of the same length in u, we have |U| 1 − |V| 1 1 or equivalently |U| 0 − |V| 0 1. A theorem of Morse and Hedlund states that a binary sequence u is periodic if and only if p(n) n for some n. A binary sequence u is called Sturmian if it is balanced and not eventually periodic. It can be shown that a binary sequence u is Sturmian if and only if it has complexity p(n) = n + 1 and is not eventually periodic. Thus, among all noneventually periodic binary sequences, Sturmian sequences are those having the smallest possible complexity.
The frequency of letter 0 (or 1) in a Sturmian sequence u = (u n ) n∈Z ∈ {0, 1} Z , defined as the limit is an irrational number. If the frequency of letter 0 in a Sturmian sequence is β, the frequency of letter 1 in that sequence is 1 − β. The following has been known (Morse & Hedlund, 1940).
Theorem 3.5 (Morse & Hedlund 1940). Let β ∈ (0, 1/2) and u ∈ {0, 1} Z . We remark that Sturmian sequences over a two-letter alphabet are also codings of trajectories of irrational initial slope in a unit square billiard obtained by labeling horizontal sides by one letter and vertical sides by the other, namely the so-called billiard sequences. Equivalently, they are also the so-

Multi-hole Sturmian system
Given a partition set Q, we can find a subsetΘ ⊆ Q, (i.e., orbits of elements ofΘ under R β are mutually disjoint) and can find integers Note that for each k one element of the set {T (k) Note also that M L ≥ 2 if L = 1. The choice of the subset Θ for a given Q is finite but not unique, whereas the choice of the integers M 1 , . . . , M L is unique. In particular, (Q) = L k=1 M k . Moreover, the cardinality ofΘ is fixed for any possible choice. We call the subsetΘ just described a least equivalent sub-partition of Q, and call the cardinality (Θ) of a least equivalent sub-partitionΘ the number of holes of the subshift (X Q,Φ , σ ) of (Φ Z , σ ).
The subsetΘ is called a 'sub-partition' because it is a subset of the partition set Q and itself can be used as a partition set provided that L 2; it is called 'equivalent' because the resulting subshift XΘ ,Θ is conjugate to X Q,Q (by Theorem 3.3); it is called 'least' because if any point is removed fromΘ, then the resulting subshift cannot be conjugate to the original one, i.e., XΘ ,Θ is not conjugate to X Q,Q ifΘ that contains zero is a proper subset ofΘ.
In fact we have the following result, which is an immediate consequence of Theorem 3.2. (iii) The system (X Q,Φ , σ ) is conjugate to (XΘ ,Θ , σ ) if it has more than one hole andΘ is a least equivalent sub-partition of Q.
An example of Corollary 3.6 is given below.
because the former has one hole, whereas the latter has two holes. Conversely, We remark that, by our construction, an L-hole Sturmian system must have a least max{2, L} symbols.
We learned that Masui (2009) constructed a partition of the unit circle similar to ourΘ here, but it requires β ∈Θ. And, a version of Theorem 4.3(i) to come in the next section in this paper was also proved in Masui (2009). The version proved there is a special case of ours when the semi-conjugacy is a conjugacy. Note that partitions similar to our Q here also appeared in Akiyama & Shirasaka (2007); Alessandri & Berthé (1998), but they did not associate their partitions with the Denjoy minimal system. The complexity of an irrational rotation sequence of partition Q = {0, q 2 , q 3 , . . . , q N } has the form p(n) = an + b with a N for n large enough. If β, q 2 , q 3 , . . . , q N are rationally independent, then a = N, b = 0 (see Alessandri & Berthé, 1998). In particular, if Q = {0, 1/2}, then p(n) = 2n for all integer n (see Rote, 1994).
The following result is an analogy of Theorem 3.4.

Coding of Denjoy minimal set
Assume that the ω-limit set ω( f ) of a Denjoy homeomorphism f satisfying Assumption A is a K-hole Cantor set. Let (Q, Φ) be a partition-symbol pair with a least equivalent sub-partitionΘ of Q. Assumẽ of open intervals A i delimited by these z i 's on S by With the given symbol set Φ = {φ 1 , φ 2 , . . . , φ N } and the intervals just constructed, define the coding for all n ∈ Z and some 1 i N. Remark that since the set {z 1 , z 2 , . . . , z N } does not intersect the ω-limit set ω( f ), the above definition is well defined. Proof.
(i) The assertion clearly holds.
(ii) Let 1 i N = (Q), q N+1 = 1, and suppose y ∈ A i . Then, by our construction, we have is not on the boundary of J ± i for every n ∈ Z and 1 i N. Thus, On the other hand, if h(x) = R n β (q i ) for some integer n and 1 i N, then Thus, by the paragraph above, it is necessary and sufficient that h( f −n (x)) ∈ J − i−1 so that E(x; Q, Φ) −n = φ i−1 , or it is necessary and sufficient that h( f −n (x)) ∈ J + i so that E(x; Q, Φ) −n = φ i . Proof.
(i) Because S is compact, f is uniformly continuous. Given any positive integer M, there exists and f n (x) ∈ A i for some 1 i N then f n (y) ∈ A i provided |n| M for any point y whose distance from x is within δ, for otherwise f n (y) ∈ S \ ω( f ). This proves the continuity.
The first equality follows. Because h is surjective, the second equality follows from Proposition 4.1(ii) and the definition (3.1). (Alternatively, the second equality can also be obtained by using (3.2).) It is known (see e.g., Katok & Hasselblatt, 1995) that the restriction of a one-hole Denjoy homeomorphism to its ω-limit set is conjugate to the restriction of the full two-shift homeomorphism to a closed invariant subset. In view of Propositions 4.1 and 4.2 we arrive at the following conclusion.
Because for any set Θ containing zero on S, there exists a Denjoy homeomorphism f of irrational rotation number β such that D(f ) coincides with O(Θ; R β ), we have an immediate corollary.
Remark 4.5 Theorem 4.3(i) says that (X Q,Φ , σ ) is always a factor of (ω( f ), f ) if the condition (4.1) holds. Of course, one could construct a partition set Q with a least equivalent sub-partition Θ such that Θ is a proper subset of Θ . Then, (ω( f ), f ) would be a factor of (X Q ,Q , σ ) via a multi-valued coding E(·; Q , Q ). The coding is multi-valued because there must be some interval in the set A whose 14 Y.-C. CHEN boundary points contain a point of (ω( f ), f ). Using a set like this A as a partition to code a Cantor set is not natural.
We elaborate a bit more on Remark 4.5 by using an example. Suppose ω( f ) is a one-hole Denjoy minimal set and Let z 1 = 1/8 ∈ [0, 1/4], z 2 = 7/12 ∈ [1/2, 2/3] and z 3 = 3/4. Then, the partition set Q leads to a set A = {(1/8, 7/12), (7/12, 3/4], (3/4, 1/8)} of intervals on S. The coding sequence of a point x ∈ ω( f ) could be defined by It is easy to see that 1/8 ∈ ω( f ), 7/12 ∈ ω( f ), but 3/4 ∈ ω( f ). Thus, the point 3/4 must be included in A (cf. (4.3) where only open intervals are used). The mapping x → E(x; Q , Q ) is 1-to-1. However, it is not continuous at x = 3/4. To see this, we could take a sequence of points in ω( f ) that converges to 3/4 from the right (clockwise). The zeroth elements of the corresponding coding sequences of these points all have the same symbol 1/2 (provided that these points are close enough to 3/4), but the zeroth element of the coding sequence of the point 3/4 is β. (Similarly, if we use (7/12, 3/4) in (4.5) and [3/4, 1/8) in (4.6) for the coding, the mapping x → E(x; Q , Q ) is still not continuous.) Furthermore, if we replace the interval in (4.6) by [3/4, 1/8), then the resulting coding sequence of x would be two-valued at x = 3/4 and at pre-images and images of 3/4 under f . If E(x; Q , Q ) is two-valued, then by identifying the two values, the mapping x → E(x; Q , Q ) would be 1-to-1 and continuous at every x in ω( f ) (with the quotient topology). This example demonstrates that the partition set Q is too 'fine' to code a one-hole Denjoy minimal set. Q is suitable for coding a two-hole Denjoy minimal set.
Theorem 5.3 (AI-limit). Assume that ω( f ) of a Denjoy homeomorphism f satisfies Assumption A. Let 0 < ε 0 < 1 be a real number, andΘ containing zero a subset of Θ. For any partition-symbol pair (Q, Φ) withΘ a least equivalent sub-partition of Q, we can construct a continuous family of Denjoy homeomorphisms f ε so that f ε 0 = f and that (ω( f ε ), f ε ) is semi-conjugate to (X Q,Φ , σ ) via a family of codings E ε (·; Q, Φ) : ω( f ε ) → X Q,Φ , which is injective if and only ifΘ = Θ, for 0 < ε ε 0 with the property: for all u ∈ X Q,Φ we have in the uniform topology.
In Theorem 5.3, (ω( f ε ), f ε ) is semi-conjugate to the (Θ)-hole Sturmian system (X Q,Φ , σ ) when ε is slightly larger than zero. As ε tends to zero from above, in the light of (5.1), (ω( f ε ), f ε ) reduces to the (Φ)-symbol (Θ)-hole Sturmian system (X Q,Φ , σ ) of partition Q. We say that the limit ε 0 is the AI-limit for the family of Denjoy homeomorphisms f ε . IfΘ = Θ, we call the AI-limit in Theorem 5.3 a nondegenerate AI-limit. Because in this situation the semi-conjugacy is in fact a conjugacy and when ε 0 the Denjoy minimal system (ω( f ε ), f ε ) reduces to a symbolic dynamical system that is conjugate to (ω( f ε ), f ε ) of small ε. If, when ε 0, a Denjoy minimal system (ω( f ε ), f ε ) reduces to a symbolic dynamical system that is not conjugate to but a factor of (ω( f ε ), f ε ) of small ε, we call such a limit a degenerate AI-limit. The limit ε 0 in Theorem 5.3 is a degenerate AI-limit if and only ifΘ = Θ.

Examples
We close this section by providing examples.
Let f be a Denjoy homeomorphism satisfying Assumption A. Let the length |I where η satisfying 0 < η+l n depend continuously on a parameter ε, then f , which has S \ 1 k K, n∈Z I (k) n as its ω-limit set, depends on ε as well. Write it as f ε .

Proofs of theorems
Proof.
(i) If the statement is not true, then s = t. Subsequently, there exists an integer l such that R l β (t) lies in the interior of J ± 1 while R l β (s) lies in the interior of J ± j for some 2 j N = (Q). Consequently, ν ± (t; Q, Φ) l = 1 = j = ν ± (s; Q, Φ) l , contradicting to the hypothesis of the proposition.
(ii) If s = t and O(s; R β ) ∩ Q = ∅, then for every integer n the orbit point R n β (s) does not locate on the boundary of J ± i for all 1 i N.
On the other hand, if ν + (s; Q, Φ) = ν − (t; Q, Φ), then s = t. (For if s = t, then it follows by the same argument used to prove (i) that there exists l ∈ Z such that ν + (s; Q, Φ) l = 1 but ν − (t; Q, Φ) l = j for some j = 1.) Suppose R m β (t) = q k for some m ∈ Z and 1 k N.
Proof. If t ∈ O(Q; R β ) then for every integer n the orbit point R n β (t) does not locate at any boundary point of J ± i for all 1 i N = (Q). Thus, ν − (t; Q, Φ) = ν + (t; Q, Φ). Given any integer M > 0 there exists δ > 0 such that for every |n| M both orbit points R n β (s) and R n β (t) lie in the same interior of intervals J + i and J − i for some i provided |s − t| < δ. This means that ν − (s; Q, Φ) n = ν − (t; Q, Φ) n = ν + (s; Q, Φ) n = ν + (t; Q, Φ) n for all |n| M, and implies the continuity at t.
If t = R m β (q j ) for some m ∈ Z and 1 j N then there are N j number of integers m 1 , m 2 , . . . , m N j (all depending on m and j) with 1 N j N and m 1 = −m for which R m 1 β (t), R m 2 β (t), . . . , R m N j β (t) ∈ Q, and R n β (t) ∈ Q for any other integer n. Therefore, none of the points in {R n β (t)|n ∈ Z \ {m 1 , . . . , m N j }} is a boundary point of J ± i for all 1 i N. Thus, for any integer M > 0 there exists δ > 0 such that for every |n| M and n ∈ {m 1 , . . . , m N j } both orbit points R n β (s) and R n β (t) lie in the same interior of both intervals J + i and J − i and that for every n ∈ {m 1 , . . . , m N j } points R n β (s) and R n β (t) lie in the same interval J + i for some 1 i N provided 0 < s − t < δ. This implies the property (6.2). Similarly, the property (6.1) can be proved. Proposition 6.3 Both the inverses ν − (t; Q, Φ) → t and ν + (t; Q, Φ) → t for any partition-symbol pair (Q, Φ) are continuous in S: suppose u ∞ , u 1 , u 2 , . . . all belong to X Q,Φ with lim n→∞ u n = u ∞ , and suppose t ∞ , t 1 , t 2 , . . . are corresponding points in S given by the injectivity of each of the mappings t → ν ± (t; Q, Φ). Then Proof. R n β (t ∞ ) ∈ Q for all integer n if u ∞ = ν + (t ∞ ; Q, Φ) = ν − (t ∞ ; Q, Φ). In this case t ∞ is contained in the interior of an interval J + j or J − j for some 1 j N = (Q). Suppose (t n ) n 1 does not converge to t ∞ . Then, it contains a subsequence that converges to another point, say,t. It follows from Proposition 6.2 that the sequence (u) n≥1 must converge either to ν + (t; Q, Φ) or to ν − (t; Q, Φ). In other word, u ∞ = ν + (t; Q, Φ) or ν − (t; Q, Φ). But, it follows from Proposition 6.1(i) that ν + (t; Q, Φ) = ν + (t ∞ ; Q, Φ) = ν − (t ∞ ; Q, Φ) = ν − (t; Q, Φ), a contradiction.

Proof of Theorem 3.2.
It is well-known that the shift σ is a homeomorphism of Φ Z . Thus, σ is also a homeomorphism of X Q,Φ if X Q,Φ is a compact invariant subset of Φ Z . Since the latter is compact, it is enough to show that X Q,Φ is invariant and closed. Because for any t ∈ S, the shift σ is a bijection of X Q,Φ and X Q,Φ is invariant under σ . Now, we show that X Q,Φ is a closed subset. Any infinite sequence of points in X Q,Φ must contain an infinite subsequence of points of the form ν + (t n ; Q, Φ) n 1 or of the form ν − (t n ; Q, Φ) n 1 . Without loss of generality, we can assume that the first case happens. Taking a subsequence if necessary, we assume that the sequence (t n ) n 1 converges to a point t ∞ by the compactness of S. Then, (t n ) n 1 contains either a subsequence that converges to t ∞ from the left (anti-clockwise) or a subsequence that converges to t ∞ from the right (clockwise). If the first case holds for (t n ) n 1 , that is, lim n→∞ t n = t − ∞ (by taking a subsequence again if necessary), then by Proposition 6.2, we infer that lim n→∞ ν + (t n ; Q, Φ) = ν − (t ∞ ; Q, Φ). If the second case holds, that is, lim n→∞ t n = t + n , then lim n→∞ ν + (t n ; Q, Φ) = ν + (t ∞ ; Q, Φ). This proves that X Q,Φ is closed. Now, X Q,Φ is a subset of the totally disconnected set Φ Z , so is itself totally disconnected. Proposition 6.2 implies that every point in X Q,Φ is a limit point of points in X Q,Φ . Since X Q,Φ is compact, it is a Cantor set.
Because O(s; R β ) is dense in S for any s ∈ S, it follows from (6.3) and (6.4) and Proposition 6.2 again that O(u; σ ) is dense in X Q,Φ for any u ∈ X Q,Φ . This completes the proof of the minimality.